The problem is that I am making this calculation thousands of time per second in a radar simulation program and if I can help it, I would like to not waste system resources on unnecessary conversions. I found the answer by the way.
For anyone else who might be looking for this solution:
The coordinates of the point dividing the line segment P1P2 in the ratio a/b are:
It looks intimidating, but it isn't. The hardest part is making sure you have the correct sign after calculating arctan(). If you are implementing it in Java, you can use:
Math.atan2(y, x) computes the phase theta by computing an arc tangent of y/x in the range of -pi to pi. In the case of being negative, just add 2*PI.
tempTheta = Math.atan2(tempRange, tempAz);
if(tempTheta < 0) tempTheta += (2 * Math.PI);
For more on Points and Lines in Polar Coordinates, look here: Math Forum: Ask Dr. Math FAQ: Polar Coordinates